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A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)). 21
1, 2, 3, 6, 2, 4, 5, 10, 15, 30, 10, 20, 3, 6, 9, 18, 6, 12, 2, 4, 6, 12, 4, 8, 7, 14, 21, 42, 14, 28, 35, 70, 105, 210, 70, 140, 21, 42, 63, 126, 42, 84, 14, 28, 42, 84, 28, 56, 5, 10, 15, 30, 10, 20, 25, 50, 75, 150, 50, 100, 15, 30, 45, 90, 30, 60, 10, 20, 30, 60, 20, 40, 3, 6, 9, 18, 6, 12, 15, 30, 45, 90, 30, 60, 9, 18, 27 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of nonzero digits that occur on the slope (k-1) levels below the "maximal slope" in the factorial base representation of n. See A275811 for the definition of the "digit slopes" in this context.

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..40320

Index entries for sequences related to factorial base representation

FORMULA

a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

Other identities and observations. For all n >= 0:

a(n) = A275735(A225901(n)).

a(A007489(n)) = A002110(n).

A001221(a(n)) = A060502(n).

A001222(a(n)) = A060130(n).

A007814(a(n)) = A260736(n).

A051903(a(n)) = A275811(n).

A048675(a(n)) = A275728(n).

A248663(a(n)) = A275808(n).

A056169(a(n)) = A275946(n).

A056170(a(n)) = A275947(n).

A275812(a(n)) = A275962(n).

EXAMPLE

For n=23 ("321" in factorial base representation, A007623), all three nonzero digits are maximal for their positions (they all occur on "maximal slope"), thus a(23) = prime(1)^3 = 2^3 = 8.

For n=29 ("1021"), there are three nonzero digits, where both 2 and the rightmost 1 are on the "maximal slope", while the most significant 1 is on the "sub-sub-sub-maximal", thus a(29) = prime(1)^2 * prime(4)^1 = 2*7 = 28.

For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the sub-maximal, and the most significant 1 is on the "sub-sub-sub-maximal", thus a(37) = prime(1) * prime(2) * prime(4) = 2*3*7 = 42.

For n=55 ("2101"), the least significant 1 is on the maximal slope, and the digits "21" at the beginning are together on the sub-sub-maximal slope (as they are both two less than the maximal digit values 4 and 3 allowed in those positions), thus a(55) = prime(1)^1 * prime(3)^2 = 2*25 = 50.

PROG

(Scheme, with memoization-macro definec)

(definec (A275734 n) (if (zero? n) 1 (* (A275732 n) (A275734 (A257684 n)))))

(Python)

from operator import mul

from sympy import prime, factorial as f

def a007623(n, p=2): return n if n<p else a007623(int(n/p), p+1)*10 + n%p

def a275732(n):

    x=str(a007623(n))[::-1]

    return 1 if n==0 or x.count("1")==0 else reduce(mul, [prime(i + 1) for i in xrange(len(x)) if x[i]=='1'])

def a257684(n):

    x=str(a007623(n))[:-1]

    y="".join([str(int(i) - 1) if int(i)>0 else '0' for i in x])[::-1]

    return 0 if n==1 else sum([int(y[i])*f(i + 1) for i in xrange(len(y))])

def a(n): return 1 if n==0 else a275732(n)*a(a257684(n))

print [a(n) for n in xrange(101)] # Indranil Ghosh, Jun 19 2017

CROSSREFS

Cf. A001221, A001222, A002110, A007489, A007814, A048675, A051903, A056169, A056170, A060130, A060502, A225901.

Cf. A257684, A275732.

Cf. A275811.

Cf. A260736, A275728, A275808, A275812, A275946, A275947, A275962.

Cf. A275804 (indices of squarefree terms), A275805 (of terms not squarefree).

Cf. also A275725, A275733, A275735, A276076 for other such prime factorization encodings of A060117/A060118-related polynomials.

Sequence in context: A225820 A153634 A224910 * A216993 A073546 A216975

Adjacent sequences:  A275731 A275732 A275733 * A275735 A275736 A275737

KEYWORD

nonn,base

AUTHOR

Antti Karttunen, Aug 08 2016

STATUS

approved

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Last modified October 16 03:37 EDT 2019. Contains 328040 sequences. (Running on oeis4.)